Understanding Linear Momentum in Collisions and Conservation Laws

1. Chapter 9Linear Momentum

2. Assignment 3Chp 6-2, 6-57, 6-629-13, 9-34, 9-54, 9-62, 9-83Due Monday 9:00 AM

3. Topics • Momentum and Its Relation to Force • Conservation of Momentum • Collisions • Conservation of Energy and Momentum in Collisions • Elastic & Inelastic Collisions in One Dimension • Elastic & Inelastic Collisions in Two Dimensions

4. Center of Mass (CM) • Center of Mass and Translational Motion • Systems of Variable Mass; Rocket Propulsion

5. Linear Momentum What is momentum and why is it important? Momentum p is the product of mass and velocity for a particle or system of particles. The product of m and v is conserved in collisions and that is why it is important. It is also a vector which means each component of momentum is conserved. It has units of kg m/s or N-s.

6. Linear Momentum form of Newton’s 2nd Law Now note we have also a force when the mass changes

7. Law of Conservation of Linear Momentum http://www.colorado.edu/physics/2000/bec/

8. Therefore each component of the momentum Px, Py, Pz is also constant. Gives three equations: Px= constant Py = constant Pz = constant If one component of the net force is not 0, then that component of momentum is not a constant. For example, consider the motion of a horizontally fired projectile. The y component of P changes while the horizontal component is fixed after the bullet is fired.

9. Conservation of Momentum Conservation of momentum holds during collisions. A collision takes a short enough time that we can ignore external forces. Since the internal forces are equal and opposite, the total momentum is constant.

10. Types of Collision • Two colliding cars. • Rubber ball bouncing off the floor • Two balls colliding on a pool table. • Tennis racquet striking a ball • A bullet striking a target. • High speed electron striking a proton

11. Conservation of Momentum Example Railroad cars collide and stick: momentum conserved ? Yes. A 10,000-kg railroad car, A, traveling at a speed of 24.0 m/s strikes an identical car, B, at rest. If the cars lock together as a result of the collision, what is their common speed immediately after the collision?

12. Explosions • Opposite of a collision • Chemical or nuclear explosion • Exploding bomb • Firing a bullet • Critical Mass

13. Conservation of Momentum Example: Rifle recoil. Calculate the recoil velocity of a 5.0-kg rifle that shoots a 0.020-kg bullet at a speed of 620 m/s.

14. F(t) What happens during a collision on a short time scale? Consider one object the projectile and the other the target.

15. J is called the impulse Change in momentum of the ball. J is a vector Alsoyou can “rectangularize” the graph

16. Shape of two objects while colliding with each other. Obeys Newtons third Law

17. Racquet and Tennis Ball Collision Figure 9.8

18. Example: Andy Rodick has been clocked at serving a tennis ball up to 149 mph(70 m/s). The time that the ball is in contact with the racquet is about 4 ms. The mass of a tennis ball is about 300 grams. • What is the average force exerted by the racquet • on the ball?

19. b) What is the acceleration of the ball? c) What distance does the racquet go through while the ball is still in contact?

20. One Dimension Head-on Collision Total momentum before = Total momentum after True for any collision

21. One Dimension Elastic Collision Kinetic energy before=Kinetic energy after True only for elastic collision

22. This is important to know because you can solve problems using two equations linear in velocity. This replaces Kinetic Energy equation. Giancoli Notation

23. Problem 9-34 Giancoli A 0.060 kg ball moving with speed 4.50 m/s has a head–on collision with a 0.090 kg ball initially moving in the same direction with speed 3.00 m/s. Determine the final speed and direction of each ball assuming an elastic collision.

24. Some simple results for a head-on collision with v2i=0

25. You want to understand 3 cases m1=m2 m2>>m1 m2<<m1

26. Balls bouncing off massive floors, we have m2 >>m1 m1 m2

27. Colliding pool balls The executive toy m2 = m1 Why don’t both balls go to the right each sharing the momentum and energy?

28. Bowling pin examplem2<<m1

30. Types of Collisions Elastic Collisions: Kinetic energy and momentum are conserved Inelastic Collision: Only P is conserved. Kinetic energy is not conserved Completely inelastic collision. Masses stick together P is conserved

31. Completely Inelastic Collision Now look at kinetic energy Note Ki not equal to Kf

32. Completely Inelastic For equal masses Kf = 1/2 Ki, we lost 50% of Ki Where did it go? It went into energy of binding the objects together, such as internal energy, rearrangement of the atoms, thermal, deformation sound, etc.

33. Collisions in Two Dimensions with particle 2 at rest

34. 2 Dimensional Elastic Collisions with particle 2 at rest Write down conservation of momentum in x and y directions separately. Two separate equations because momentum is a vector. Write down conservation of kinetic energy equation. 3 equations in 2 dimensions

36. Problem 9-88

37. Problem 90-88 A bowling ball travelling at 13.0 m/s has 5 times the mass of a pin and the pin goes off at 75 degrees. Find the speed of the pin, speed of ball, and the angle of the ball.

38. Improving your Billiards

39. How do you know what angle the billiard ball deflects in an elastic collision? Head on Arbitrary Angle 90 Degrees Line of Action

40. Pool shot pocket 1 2 Assuming no spin Assuming elastic collision

41. Where do you aim the Bank shot to make the pocket? d Assuming no english on the balls Assuming elastic collision and no bank deformation d

42. How high does a ball bounce up after an almost elastic collision between floor and bouncing ball? hi -vi vf After bounce Before bounce Initial hf

43. Measuring velocities and heights of balls bouncing from a infinitely massive hard floor Almost elastic collision Almost inelastic collision

44. -v -v -v v v1f v Two moving colliding objects: HRW Problem 45 ed 6 Show demo comparing the height of a ball bouncing off the floor compared to one bouncing off another ball. Explain. m1 m2 Just after the little ball bounced off the big ball Just after the big ball bounced off the floor Just before each hits the floor m1 m2 How high does superball go compared to dropping it off the floor?

45. -v -v -v v v1f v m1 m2 Just after the big ball bounced off the floor Just after the little ball bounced off the big ball Just before each hits the floor m1 m2

46. -v v V1f=2V V2f =0 For m2 =3m1 v1f = 8/4V =2V superball has twice as much speed. 4 times higher

47. -v v V1f=3V V2f = -V For maximum height consider m2 >>m1 How high does it go? 9 times higher How about 3 balls?

48. Center of Mass

49. Center of mass (special point in a body) • Why is it important? For any rigid body the motion of the body is given by the motion of the cm and the motion of the body around the cm. • The motion of the cm is as though all of the mass were concentrated there and all external forces were applied there. Hence, the motion is parabolic like a point projectile. • How do you show projectile motion is parabolic?

50. What happens to the ballet dancers head when she raises her arms at the peak of her jump? Note location of cm relative To her waist. Her waist is lowered at the peak of the jump. How can that happen?